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Regularity of Powers of Edge Ideals: from Local Properties to Global Bounds

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Regularity of Powers of Edge Ideals: from Local Properties to Global Bounds REGULARITY OF POWERS OF EDGE IDEALS: FROM LOCAL PROPERTIES TO GLOBAL BOUNDS ARINDAM BANERJEE, SELVI KARA BEYARSLAN, AND HUY TAI` HA` Abstract. Let I = I(G) be the edge ideal of a graph G. We give various general upper bounds for the regularity function reg Is, for s ≥ 1, addressing a conjecture made by the authors and Alilooee. When G is a gap-free graph and locally of regularity 2, we show that reg Is = 2s for all s ≥ 2. This is a weaker version of a conjecture of Nevo and Peeva. Our method is to investigate the regularity function reg Is, for s ≥ 1, via local information of I. 1. Introduction During the last few decades, studying the regularity of powers of homogeneous ideals has evolved to be a central research topic in algebraic geometry and commutative alge- bra. This research program began with a celebrated theorem, proved independently by Cutkosky-Herzog-Trung [9] and Kodiyalam [24], which stated that for a homogeneous ideal I in a standard graded algebra over a field, the regularity function reg Is is asymptotically a linear function (see also [3, 33]). Though despite much effort from many researchers, this asymptotic linear function is far from being well understood. In this paper, we investigate this regularity function for edge ideals of graphs. We shall explore several classes of graphs for which this regularity function can be explicitly described or bounded in terms of combi- natorial data of the graphs. This problem has been studied recently by many authors (cf. [1, 2, 4, 5, 6, 12, 13, 21, 22, 23, 27, 30]). Our initial motivation for this paper is the general belief that global conclusions often could be derived from local information. Particularly, local conditions on an edge ideal I (i.e., conditions on reg(I : x), for x 2 V (G)) should give a global understanding of the function reg Is, for s ≥ 1. Our motivation furthermore comes from the following conjectures (see [5, 28, 29]), which provide a general upper bound for the regularity function of edge ideals, and describe a special class of edge ideals whose powers (at least 2) all have linear arXiv:1805.01434v3 [math.AC] 14 Apr 2019 resolutions. Conjecture A (Alilooee-Banerjee-Beyarslan-H`a).Let G be a simple graph with edge ideal I = I(G). For any s ≥ 1, we have reg Is ≤ 2s + reg I − 2: Conjecture B (Nevo-Peeva). Let G be a simple graph with edge ideal I = I(G). Suppose that G is gap-free and reg I = 3. Then, for all s ≥ 2, we have reg Is = 2s: 1 Our aim is to investigate Conjectures A and B using the local-global principle. Finding general upper bounds for reg I(G)s has received a special interest and generated a large number of papers during the last few years. This partly thanks to a general lower bound for reg I(G)s given in [6]; particularly, if ν(G) denotes the induced matching number of G then, for any s ≥ 1, we have reg I(G)s ≥ 2s + ν(G) − 1: (1.1) Our first main result gives a weaker general upper bound for reg I(G)s than that of Conjecture A. The motivation of this result comes from an upper bound for the regularity of I(G) given by Adam Van Tuyl and the last author, namely reg I(G) ≤ β(G) + 1, where β(G) denotes the matching number of G (see [16]). We prove the following theorem. Theorem 3.4. Let G be a graph with edge ideal I = I(G), and let β(G) be its matching number. Then, for all s ≥ 1, we have reg Is ≤ 2s + β(G) − 1: As a consequence of Theorem 3.4, for the class of Cameron-Walker graphs, where ν(G) = β(G), we have reg Is = 2s + ν(G) − 1 8 s ≥ 1: A graph G is said to be locally of regularity at most r − 1 if reg(I(G): x) ≤ r − 1 for all vertex x in G. Note that, by [8, Proposition 4.9], if G is locally of regularity at most r − 1 then reg I(G) ≤ r. In the local-global spirit, we reformulate Conjecture A to a slightly weaker conjecture as follows. Conjecture A0. Let G be a simple graph with edge ideal I = I(G). Suppose that G is locally of regularity at most r − 1, for some r ≥ 2. Then, for any s ≥ 1, we have reg Is ≤ 2s + r − 2: Our next main result proves Conjecture A0 for gap-free graphs. Theorem 4.2. Let G be a simple graph with edge ideal I = I(G). Suppose that G is gap-free and locally of regularity at most r − 1, for some r ≥ 2. Then, for any s ≥ 1, we have reg Is ≤ 2s + r − 2: It is an easy observation that if I(G)s has a linear resolution for some s ≥ 1 then G must be gap-free. Conjecture B serves as a converse statement to this observation, and has remained intractable. By applying the local-global principle, we prove a weaker statement, in which the condition reg I = 3 is replaced by the condition that G is locally linear (i.e., locally of regularity at most 2). Our main result toward Conjecture B is stated as follows. Theorem 4.5. Let G be a simple graph with edge ideal I = I(G). Suppose that G is gap-free and locally linear. Then for all s ≥ 2, we have reg Is = 2s: 2 As a consequence of Theorem 4.5, we quickly recover a result of Banerjee, which showed that if G is gap-free and cricket-free then I(G)s has a linear resolution for all s ≥ 2 (see Corollary 4.6). We end the paper by exhibiting an evidence for Conjecture A0 at the first nontrivial value of s, i.e., s = 2, for all graphs. Theorem 5.1. Let G be a graph with edge ideal I = I(G). Suppose that G is locally of regularity at most r − 1. Then, for any edge e 2 E(G), reg(I2 : e) ≤ r: Particularly, this implies that reg(I2) ≤ r + 2: Our paper is structured as follows. In the next section we give necessary notation and terminology. The reader who is familiar with previous work in this research area may want to proceed directly to Section 3. In Section 3, we discuss general upper bound for the regularity function, aiming toward Conjecture A. Theorem 3.4 is proved in this section. In Section 4, we focus further on gap-free graphs, investigating both Conjectures A0 and B using the local-global principle. Theorems 4.2 and 4.5 are proved in this section. We end the paper with Section 5, proving Theorem 5.1 and discussing briefly how an effective bound on the regularity of I(G)2 may give us information on the regularity of the second symbolic power I(G)(2). This gives a glimpse into future work on the regularity function of symbolic powers of edge ideals. Acknowledgement. Part of this work was done while the first named and the last named authors were visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). We would like to express our gratitude toward VIASM for its support and hospitality. The last named author is partially supported by Simons Foundation (grant #279786) and Louisiana Board of Regents (grant #LEQSF(2017-19)-ENH-TR-25). The authors thank Thanh Vu for pointing out a mistake in our first version of the paper. 2. Preliminaries In this section, we collect notations and terminology used in the paper. For unexplained notions, we refer the reader to standard texts [7, 11, 18, 26, 31, 34]. Graph Theory. Throughout the paper, G shall denote a finite simple graph with vertex set V (G) and edge set E(G). A subgraph G0 of G is called induced if for any two vertices 0 0 u; v in G , uv 2 E(G ) , uv 2 E(G). For a subset W ⊆ V (G), we shall denote by GW the induced subgraph of G over the vertices in W , and denote by G − W the induced subgraph of G on V (G) n W . When W = fwg consists of a single vertex, we also write G − w for G − fwg. The complement of a graph G, denoted by Gc, is the graph on the same vertex set V (G) in which uv 2 E(Gc) , uv 62 E(G). Definition 2.1. Let G be a graph. (1)A walk in G is a sequence of (not necessarily distinct) vertices x1; x2; : : : ; xn such that xixi+1 is an edge for all i = 1; 2; : : : ; n: A circuit is a closed walk (i.e., when x1 ≡ xn). (2)A path in G is a walk whose vertices are distinct (except possibly the first and the last vertices). 3 (3)A cycle in G is a closed path. A cycle consisting of n distinct vertices is called an n-cycle and often denoted by Cn: (4) An anticycle is the complement of a cycle. A graph in which there is no induced cycle of length greater than 3 is called a chordal graph. A graph whose complement is chordal is called a co-chordal graph. Definition 2.2. Let G be a graph. (1)A matching in G is a collection of disjoint edges. The matching number of G, denoted by β(G) is the maximum size of a matching in G.
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